If S.P. of an article is doubled then value of its loss % will convert to profit% . Find loss% of article.

Answer

To solve the problem step by step, we need to find the loss percentage of an article given that if the selling price (S.P.) is doubled, the loss percentage converts to a profit percentage.

### Step 1: Define the terms
Let the Cost Price (C.P.) of the article be \( x \) and the Selling Price (S.P.) be \( y \).

### Step 2: Express the loss
The loss incurred when selling the article is given by:
\[ \text{Loss} = \text{C.P.} - \text{S.P.} = x - y \]

### Step 3: Calculate the loss percentage
The loss percentage is calculated as:
\[ \text{Loss \%} = \left(\frac{\text{Loss}}{\text{C.P.}}\right) \times 100 = \left(\frac{x - y}{x}\right) \times 100 \]

### Step 4: Express the new selling price when doubled
If the selling price is doubled, the new selling price becomes:
\[ \text{New S.P.} = 2y \]

### Step 5: Calculate the new profit
The profit when the selling price is doubled is:
\[ \text{Profit} = \text{New S.P.} - \text{C.P.} = 2y - x \]

### Step 6: Set the loss percentage equal to the profit percentage
According to the problem, the loss percentage becomes the profit percentage when the S.P. is doubled. Thus, we can set up the equation:
\[ \left(\frac{x - y}{x}\right) \times 100 = \left(\frac{2y - x}{x}\right) \times 100 \]

### Step 7: Simplify the equation
We can cancel out 100 from both sides:
\[ \frac{x - y}{x} = \frac{2y - x}{x} \]
Cross-multiplying gives:
\[ x - y = 2y - x \]

### Step 8: Rearranging the equation
Rearranging the equation leads to:
\[ x + x = 2y + y \]
\[ 2x = 3y \]
Thus, we can express \( \frac{x}{y} \) as:
\[ \frac{x}{y} = \frac{3}{2} \]

### Step 9: Find the loss percentage
From the ratio \( \frac{x}{y} = \frac{3}{2} \), we can express \( y \) in terms of \( x \):
\[ y = \frac{2}{3}x \]
Now substituting \( y \) back into the loss formula:
\[ \text{Loss} = x - y = x - \frac{2}{3}x = \frac{1}{3}x \]
Calculating the loss percentage:
\[ \text{Loss \%} = \left(\frac{\frac{1}{3}x}{x}\right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer
The loss percentage of the article is:
\[ \text{Loss \%} = 33 \frac{1}{3} \% \]

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